snowleopard | 加速技术之七种武器 -

Proofs by induction are motivated by recursive definitions. So to know what to apply induction to, look for Fixpoints.

Your Fixpoints most likely work on terms indexed by single type variables Select F A, that's exactly where you want to use induction, not at the toplevel of the goal.

A Fixpoint on terms indexed by function types A -> B is useless since no subterms of any Select term are indexed by function types. For the same reason, induction is useless on such terms.

Here I think the strong type discipline actually forces you to work everything out on paper before trying to do anything in Coq (which is a good thing in my opinion). Try to do the proof on paper, without even worrying about types; explicitly write down the predicate(s) you want to prove by induction. Here's a template to see what I mean:

By induction on u, we will show